(Non)-hyperuniformity of perturbed lattices

TL;DR

This paper investigates how random perturbations affect the hyperuniformity of stationary lattices across different dimensions, establishing conditions for preservation or loss of hyperuniformity.

Contribution

It provides sharp criteria for hyperuniformity preservation under perturbations in low dimensions and constructs examples in higher dimensions where hyperuniformity is lost.

Findings

Hyperuniformity is preserved in 1D and 2D with finite $d$-moment perturbations.

In dimensions 3 and higher, small perturbations can destroy hyperuniformity.

Existence of hyperuniform processes with very slow number variance decay.

Abstract

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.